6.3 Wave-map structure

It is of interest to note that Equations (16) – (17) are of wave-map type. The domain is given by
with the metric

and the target is with the metric

Equations (16) – (17) are the wave-map equations for a map from to , which is
independent of the -coordinate. In other words, they coincide with the Euler–Lagrange equations
corresponding to the action

6.3.1 Representations of hyperbolic space

It is of interest to note that is isometric to hyperbolic space. In fact,

gives an isometry from to the upper half plane model of hyperbolic space, , where
and

For certain considerations, the upper half plane is an inappropriate model of hyperbolic space. The reason is
that the upper half plane has a preferred boundary point, namely infinity; see Section 9.3. The disc model,
given by the Riemannian metric

on the open unit disc does not have preferred boundary points, and therefore is sometimes preferable.
Using complex notation,

yields an isometry from the upper half plane to the disc model. By composing with , we
obtain an isometry from to . The expression is given by